Suppose that z1, z2, z3 are three vertices of an equilateral tria

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 Multiple Choice QuestionsMultiple Choice Questions

61.

If (2 + i) and 5 - 2i are the roots of the equation (x2 + ax + b )(x2 + ex + d) = 0, where a, b, c and d are real constants, then product of all the roots of the equation is

  • 40

  • 95

  • 45

  • 35


62.

Which of the following is /are always false?

  • A quadratic equation with rational coefficients has zero or two irrational roots

  • A quadratic equation with real coefficients has zero or two non-real roots

  • A quadratic equation with irrational coefficients has zero or two irrational roots

  • A quadratic equation with integer coefficients has zero or two irrational roots


63.

The number of solution(s) of the equation x + 1 - x - 1 = 4x - 1 is/are

  • 2

  • 0

  • 3

  • 1


64.

The value of z2 + z - 32 + z - i2 is minimum when z equals

  • 2 - 23i

  • 45 + 3i

  • 1 + i3

  • 1 - i3


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65.

The solution of the equation log101log7x + 7 + x = 0 is

  • 3

  • 7

  • 9

  • 49


66.

In a ABCtanA and tanB are the roots of pq(x2 + 1) = r2x. Then, ABC is

  • a right angled triangle

  • an acute angled triangle

  • an obtuse angled triangle

  • an equilateral triangle


67.

Let f(x) = 2x+ 5x + 1. If we write f(x) as f(x) = a(x + 1)(x - 2) + b(x - 2)(x - 1) + c(x - 1)(x + 1) for real numbers a, b, c then

  • there are infinite number of choices for a, b, c

  • only one choice for a but infinite number of choices for b and c

  • exactly one choice for each of a, b, c

  • more than one but finite number of choices for a, b, c


68.

If α, β are the roots of ax2 + bx + c = 0 (a  0) and α + h, β + h are the roots of px2 + qx + r = 0 (p  0), then the ratio of the squares of their discriminants is

  • a2 : p2

  • a : p2

  • a2 : p

  • a : 2p


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69.

Suppose that z1, z2, z3 are three vertices of an equilateral triangle in the Argand plane. Let α = 123 + i and β be a non-zero complex number. The points αz1 + β, αz2 + β, αz3 + β will be

  • the vertices of an equilateral triangle

  • the vertices of an isosceles triangle 

  • collinear

  • the vertices of a scalene triangle


A.

the vertices of an equilateral triangle

z1 - z2 = z2 - z3               = z3 - z1 = k Also, α = 123 + i α = 123 + 1             = 12 × 2             =  1

Let A = αz1 + β, B = αz2 + β and C = αz3 + βNow, AB = αz2 + β - αz1 + β                 = αz2 - z1                 = α  z2 - z1                 = 1  z2 - z1                 = z2 - z1 = k

Similarly, BC = CA = k

Hence, the points  αz1 + β, αz2 + β and αz3 + β are the vertices of an equilateral triangle.


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70.

In the Argand plane, the distinct roots of 1 + z + z3 + z4 = 0 (z is a complex number) represent vertices of

  • a square

  • an equilateral triangle

  • a rhombus

  • a rectangle


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